Fel's Conjecture: Number Theory & Semigroups

• Fel's Conjecture is a dual conjecture; in additive number theory, it posits that for even integers, if all prime-shifted differences are composite then at least one must have a large prime divisor, implying Goldbach's conjecture.
• It uses a controlled sequence of prime differences and an inductive sieving process, leveraging combinatorial constraints and Bertrand’s postulate to achieve a minimalist proof strategy.
• In numerical semigroups, the conjecture provides an explicit formula linking gap statistics and generator invariants to alternating syzygy power sums, and has been formalized using machine-assisted proof techniques.

Fel's Conjecture refers to distinct conjectures in two separate areas of mathematics, both noted for their connections to deep structural problems and for their precise, elementary formulations. The first and classical instance concerns additive number theory—specifically, a conjecture on "prime-shifted" composites whose validity would imply Goldbach's conjecture (Williamson, 2018). The second and more recent instance addresses syzygies of numerical semigroups, proposing a universal explicit formula for alternating syzygy power sums in terms of gap statistics and generator invariants (Chen et al., 3 Feb 2026).

1. Formulation in Additive Number Theory

Let $p_1, p_2, \dots, p_k$ be the first $k$ odd primes in increasing order, and let $n$ be an even integer with $n > p_k$. Fel's Conjecture states:

$$\text{If for every}\ 1 \leq i \leq k\ \text{the integer}\ n-p_i\ \text{is composite, then there exists some}\ 1 \leq i \leq k\ \text{and a prime divisor}\ q \mid (n-p_i)\ \text{with}\ q \geq p_k.$$

In symbolic form: $$\left(\forall i \leq k,\, n-p_i \notin \mathbb{P}\right) \implies \left(\exists i \leq k,\, \exists \text{ prime } q \mid (n-p_i) \text{ with } q \geq p_k\right).$$

This statement is a weakening of Grimm’s conjecture applied to a strictly-structured sequence of differences, tailored so that at least one of the $n-p_i$ has a "large" (i.e., $\geq p_k$) prime factor.

2. Deduction of Goldbach's Conjecture

Fel's Conjecture immediately yields a proof of Goldbach’s conjecture via an inductive "sieving" process:

• If for some $i \leq k$, $n-p_i$ is prime, $n$ is expressed as the sum of two primes.
• Otherwise, the conjecture ensures that some $n-p_i$ admits a prime factor $q \geq p_k$; if $q > p_k$, one increments $k$ and repeats. If $q = p_k$, combinatorial constraints and postulates (e.g., Bertrand's postulate) guarantee a strictly increasing lower bound $p_{k+1} < n$, allowing further iteration.
• Since there are only finitely many primes less than $n$, the process concludes with a representation of $n$ as a sum of two primes.

3. Examples and Computational Evidence

For small $k$, explicit computations exemplify the conjecture's mechanism:

• For $k=1$ ($p_1=3$, $n>3$ even), if $n-3$ is composite, it is a multiple of $3$ (the only allowable prime $q=p_1$).
• For $k=2$ ($p_1=3,\ p_2=5$, $n>5$ even), the first nontrivial case is $n=30,\ n-3=27=3^3,\ n-5=25=5^2$; $n-5$ yields $q=5=p_2$ as required.
• An infinite family with $k=1$ and $n=3^r+3$ ($r>1$) always satisfies the conjecture with $q=p_1$.

Extensive computer verification for large ranges of $k$ and $n$ supports the universal validity of the conjecture, with the "large" prime divisor typically associated with a small index $i$ (relative to $k$), and equality $q=p_k$ observed only in special, highly structured cases (Williamson, 2018).

4. Connections to Broader Additive Problems

Fel’s Conjecture situates itself as a generalization of Grimm’s conjecture, shifting from consecutive composites to a controlled sequence linked to prime increments. Its significance lies in its reduction of the Goldbach problem to a divisor-distribution assertion—the difficulty of which is concentrated in understanding when all shifts $n-p_i$ fail to be prime yet fail to exhibit a sufficiently large distinct factor.

Classical sieve methods or the theory of primes in arithmetic progressions might be adaptable but require bounding the occurrence of small prime divisors within the given family of $n-p_i$. The conjecture encourages further exploration within analytic and combinatorial sieve frameworks.

5. Fel's Conjecture in the Theory of Numerical Semigroups

A distinct conjecture, also termed Fel’s Conjecture in the literature, arises in the context of numerical semigroups and their syzygies (Chen et al., 3 Feb 2026).

Let $S = \langle d_1, \dots, d_m \rangle$ be a numerical semigroup with $\gcd(d_1,\dots,d_m)=1$, and let $k[S]$ be its semigroup ring over a field $k$, with Hilbert numerator $Q_S(z)$. For $p \geq 0$, Fel's Conjecture asserts the following explicit formula for the normalized alternating syzygy power sums,

$$K_p(S) = \sum_{r=0}^{p} \binom{p}{r} T_{p-r}\bigl(\sigma_1,\dots,\sigma_{p-r}\bigr) G_r(S) + \frac{2^{p+1}}{p+1} T_{p+1}(\delta_1, \dots, \delta_{p+1}),$$

where $G_r(S) = \sum_{g \notin S} g^r$ are the gap power sums, and $T_n$ are universal symmetric polynomials evaluated at the generator sums $\sigma_k = \sum_i d_i^k$, $\delta_k = (\sigma_k - 1)/2^k$.

This formula reduces syzygy power sums to explicit, universal combinations of gap statistics and generator statistics, utilizing coefficient extraction from exponential generating functions associated with $S$ (Chen et al., 3 Feb 2026).

6. Proofs and Formalization

The proof of the numerical semigroup instance proceeds by:

• Expressing $\mathcal{C}_n(S)$ (the alternating power sums of syzygy degrees) as coefficients in the expansion of $1-Q_S(z)$.
• Relating $Q_S(z)$ and the generating series for gaps, $\Phi_S(z)$, through manipulation of the Hilbert series $H_S(z)$ and denominator $P_S(z)$.
• Translating the explicit formula into a generating function equality, and extracting the coefficients via established combinatorial identities for $T_n$.
• The proof has been formalized in the Lean/Mathlib ecosystem via AxiomProver, demonstrating the conjecture's alignment with machine-verifiable mathematics at a high level of abstraction.

7. Impact and Outlook

Fel’s Conjecture in additive number theory provides a minimalist route to the Goldbach conjecture, redirecting the analytic challenge to the distributions of prime divisors in controlled sequences. Its empirical confirmation for extensive ranges underlines the apparent robustness of the underlying principle. In commutative algebra, Fel’s explicit formula for syzygy power sums in numerical semigroups unifies combinatorial, algebraic, and analytical invariants, providing a benchmark example for the power of generating function methods and formal proof technology.

In both domains, Fel’s Conjecture exemplifies the power of elementary, explicitly checkable statements in redirecting the flow of deep mathematical questions and the potential of machine-assisted proof verification for complex algebraic identities (Williamson, 2018, Chen et al., 3 Feb 2026).